\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{exercise in groupoidification - the path integral}

\begin{quote}%
under construction


\end{quote}
On the formalization of the process of [[quantization]] -- by [[category theory|abstract nonsense]] -- from classical [[∞-model]] data to the corresponding [[quantum field theory]] .

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{background_and_motivation}{Background and motivation}\dotfill \pageref*{background_and_motivation} \linebreak
\noindent\hyperlink{general_ambient_structure}{General ambient structure}\dotfill \pageref*{general_ambient_structure} \linebreak
\noindent\hyperlink{the_charged_particle}{The charged particle}\dotfill \pageref*{the_charged_particle} \linebreak
\noindent\hyperlink{target_space_and_background_field}{Target space and background field}\dotfill \pageref*{target_space_and_background_field} \linebreak
\noindent\hyperlink{parameter_space_and_kinetic_action}{Parameter space and kinetic action}\dotfill \pageref*{parameter_space_and_kinetic_action} \linebreak
\noindent\hyperlink{the_quantization}{The quantization}\dotfill \pageref*{the_quantization} \linebreak


\hypertarget{background_and_motivation}{}\subsection*{{Background and motivation}}\label{background_and_motivation}

The search is on for the abstract formalization of the process of \emph{[[quantization]]} -- the process that reads in a ``[[classical field theory]]'' -- for instance presented in form of a [[gauge theory]] or in form of a [[∞-model]] background field data -- and spits out the corresponding [[quantum field theory]].

There is an open problem of mathematical (-[[physics]]) model building:

\begin{itemize}%
\item what is the true formalism behind [[quantization]]?

\end{itemize}
One that is to quantization as, say, [[symplectic geometry]] is to [[Hamiltonian mechanics]]?

The formalism of [[FQFT]] clearly suggests that the fundamental description of [[quantization]] is some natural operation on higher functors.

While for various aspects and facets of this question there are well-developed formalisms -- such as [[geometric quantization]] or [[deformation quantization]] or [[BV-BRST formalism]] -- a full answer is certainly still missing, not the least because the full formalization of the \emph{question} itself has still to be established.

Considerable progress on this formulation of the question has been achieved with the formalization and proof of the [[cobordism hypothesis]] in \emph{[[On the Classification of Topological Field Theories]]} by [[Jacob Lurie]]. This at least indicates what the \emph{result} of any full quantization procedure should be in that it clarifies what exactly a [[TQFT]] [[FQFT]] is: a morphism from the [[(∞,n)-category of cobordisms]] $Z : Bord_n \to C$.

In \emph{[[On the Classification of Topological Field Theories]]} Jacob Lurie indicates some first towards finding a similar formalization of ``classical field theory'' (in terms of his $(\infty,n)$-categories of ``families'') and a systematic procedure for turning the classical theory into the quantum theory. These thoughts were further developed in the article \emph{[[Topological Quantum Field Theories from Compact Lie Groups]]} . But for the moment, that, too, remains a bit sketchy.

For the purpose of the present entry this indication of a quantizaton proposal by Lurie et al. mainly serves as a reference for the idea itself that a formalization of something interesting is to be sought here, and of the kind of [[category theory|abstract nonsense]] answer one hopes to find. We will however discuss a somewhat different-looking approach. It may well be related to the Lurie-et al proposal in the end, but for the time being we shall not concentrate on that relation.

Rather, the approach for a formalization of the quantization procedure that shall be discussed at this entry here draws from a few different sources:

\begin{enumerate}%
\item The observation that a classical background field that should serve as the input for a quantization of a $\sigma$-model that describes the dynamics of an object charged under this field is encoded by differential cocycles as as described at [[schreiber:differential cohomology in an (∞,1)-topos]].


\item The idea that by applying a pull-push quantization prescription a differential cocycle on a target space $X$ gives rise to a differential cocycle on a \emph{parameter space} $\Sigma$, which may be thought of as one of the bordisms appearing in the [[FQFT]]-description of quantum field theory.

The pull-push operation here is akin to that in [[geometric ∞-function theory]], where a [[quantum field theory]] is obtained from a [[∞-model]] target space object $X$ by homming [[extended cobordism]] [[cospan]]s $\Sigma_in \to \Sigma \leftarrow \Sigma_{out}$ into the target object and then pull-pushing [[geometric function object]]s through the resulting [[span]]s of configuration space objects $[\Sigma_{in},X] \leftarrow [\Sigma,X] \to [\Sigma_{out},X]$.

The main result of David Ben-Zvi et. al.`s work on this approach is that they point out that as soon as the [[geometric function object]] one uses satisfies the two [[geometric ∞-function theory|fundamental theorems of geometric infinity-function theory]], a considerable amount of rich structure that has in parts been known by itself gets unified into one coherent elegant story: the nature of partition functions (i.e. traces), of centers, of Hochschild (co)homology, Deligne-Kontsevich-statements, etc. all are understood by means of a suitable [[geometric function theory]] as induced from the underlying geometry of configuration space objects $[\Sigma,X]$ as well as the [[loop space object]]s of $X$.

The resulting pull-push operation is an example or a generalization of what [[John Baez]] discusses under the term [[groupoidification]].


\item The observation that a differential coccycle on a [[Lorentzian manifold]] $\Sigma$ gives rise to a [[local net]] of observables, as used in the formalizaton of QFT known as [[AQFT]]. (As described \href{http://ncatlab.org/schreiber/files/AQFTfromFQFT.pdf}{here}).

So the procedure discussed here regards differential cocycles on target space as classicai field theories, regards their quantization as a way to obtain a differential cocycle on Lorentzian parameter space, and identifies this as a quantum field theory by associating a local net of observables to it.

These local nets, in turn, are akin to [[factorization algebra]]s, which in the Euclidean (meaning non-Lorentzian setting) relate back to cobordism representations via the notion of [[topological chiral homology]]. However the -- physically crucial -- Lorentzian structure invoked here is not otherwise considered in these functorial axiomatization of quantum field theory.



\end{enumerate}
\hypertarget{general_ambient_structure}{}\subsection*{{General ambient structure}}\label{general_ambient_structure}

The ambient context is the [[(∞,1)-topos]] $\mathbf{H} := Sh_{(\infty,1)}(CartSp)$ of [[Lie ∞-groupoid]]s and the $(\infty,2)$-topos of smooth $(\infty,1)$-categories, which we model, respectively, as the [[Bousfield localization of model categories|left Bousfield localization]] of the [[model structure on functors]] $[CartSp^{op}, sSet_{Quillen}]$ and $[CartSp^{op}, sSet^+]$, respectively, where

\begin{itemize}%
\item [[CartSp]] is the site of Cartesian spaces,


\item $sSet_{Quillen}$ the standard [[model structure on simplicial sets]];


\item and $sSet^+$ the [[model structure for Cartesian fibrations]] over the points, hence the [[simplicial model category]] for quaasi-categories.



\end{itemize}
For the definition of the [[schreiber:path ∞-groupoid]] functor $\mathbf{\Pi}$ and the induced theory of [[schreiber:differential cohomology in an (∞,1)-topos]], we make use of the discussion at [[schreiber:differential cohomology in an (∞,1)-topos -- survey]].

When working with fibrant objects in the model, we will frequently use the constructions and notation from [[category of fibrant objects]]. Notably for $\mathbf{B}G$ a fibrant [[delooping]] object in the model we write $(\mathbf{B}G)^I$ for the [[path object]] we write $\mathbf{E}G$ for the [[pullback]]

\begin{displaymath}
\itexarray{
    \mathbf{E}G &\to& *
    \\
    \downarrow && \downarrow
    \\
    (\mathbf{B}G)^I &\stackrel{d_1}{\to}& \mathbf{B}G
  }
\end{displaymath}
and $\mathbf{E}G \to \mathbf{B}G$ for the remaining map, induced from $d_0 : (\mathbf{B}G)^I \to \mathbf{B}G$.

\hypertarget{the_charged_particle}{}\subsection*{{The charged particle}}\label{the_charged_particle}

We describe the general theory for the simple example of the charged particle.

\hypertarget{target_space_and_background_field}{}\subsubsection*{{Target space and background field}}\label{target_space_and_background_field}

The background field for the charged particle that we want to consider is the [[electromagnetic field]]. The data involved is

\begin{itemize}%
\item the \textbf{target space} $X$ -- a smooth [[manifold]];


\item the \textbf{structure group} or \textbf{gauge group} $G = U(1)$;


\item a choice of representation

\begin{displaymath}
\rho : \mathbf{B}G \to Vect_{\mathbb{C}}
  \,,
\end{displaymath}
taken to be the canonical representation on $V = \mathbb{C}$;


\item the \textbf{background field} given by a

\begin{itemize}%
\item a $U(1)$-[[principal bundle]] $P \to X$ classified by a [[cocycle]] $g : X \to \mathbf{B}U(1)$ in $\mathbf{H}$ which in the model is given by an [[anafunctor]] $X \stackrel{\simeq}{\leftarrow} Y \to \mathbf{B}U(1)$;


\item a [[connection on a bundle|connection]] $\nabla$ on this bundle, which in the model is given by a diagram

\begin{displaymath}
\itexarray{
  Y &\stackrel{g}{\to}& \mathbf{B}U(1)
  \\
  \downarrow && \downarrow
  \\
  \mathbf{\Pi}(Y) &\stackrel{\nabla}{\to}& 
  \mathbf{E}\mathbf{B}U(1)
  }
  \,,
\end{displaymath}
and whose [[field strength]] is given by the composite

\begin{displaymath}
F : \mathbf{\Pi}(Y) \stackrel{\nabla}{\to}
  \mathbf{E}\mathbf{B}U(1) \to \mathbf{B}^2 U(1)
  \,.
\end{displaymath}


\end{itemize}


\end{itemize}
\hypertarget{parameter_space_and_kinetic_action}{}\subsubsection*{{Parameter space and kinetic action}}\label{parameter_space_and_kinetic_action}

Let $\Sigma = \mathbb{R}$ be the \textbf{parameter space}, the \textbf{worldline}, regarded as a [[Lorentzian manifold]] and write $\mathbf{\Pi}(\Sigma)$ for the \href{http://ncatlab.org/nlab/show/smooth+Lorentzian+space#PathnCategory}{Lorentzian path category}.

To define the kinetic action, we first form the $\rho$-associated background field

\begin{displaymath}
\itexarray{
    X &\stackrel{g}{\to}& \mathbf{B}U(1) &\stackrel{\rho}{\to}& Vect
    \\
    \downarrow && \downarrow && \downarrow
    \\
    \mathbf{\Pi}(X) &\stackrel{\nabla}{\to}& \mathbf{E} \mathbf{B}U(1) 
      &\stackrel{}{\to}& \mathbf{E}\mathbf{B}U(1) \coprod_{\mathbf{B}U(1)} Vect
  }
\end{displaymath}
and then pull this back to the [[multisymplectic geometry|extended configuration space]] $X \times \Sigma$.

The morphisms in the [[product]] category $\mathbf{\Pi}(X) \times \mathbf{\Pi}(\Sigma)$ are paths $\gamma_X : [0,1] \to X$ in $X$ on whose base we have a (pseudo)[[Riemannian metric]], which is the pullback of the metric $\mu_\Sigma$ on $\Sigma$ along $\gamma_\Sigma : [0,1] \to \Sigma$. We can consider the \emph{kinetic action} to be a differential cocycle

\begin{displaymath}
\itexarray{
    \Sigma \times X 
     &\stackrel{}{\to}& \mathbf{B}U(1) &\stackrel{}{\to}& Vect
    \\
    \downarrow && \downarrow && \downarrow
    \\
    \mathbf{\Pi}(\Sigma) \times 
     \mathbf{\Pi}(X) &\to& \mathbf{E} \mathbf{B}U(1) 
      &\stackrel{}{\to}& \mathbf{E}\mathbf{B}U(1) 
      \coprod_{\mathbf{B}U(1)} Vect
  }
\end{displaymath}
which sends the path $\gamma_X : [0,1] \to X$ of parameter length $\gamma_\Sigma : [0,1] \to \Sigma$ to

\begin{displaymath}
\mathbb{C}
    \stackrel{\exp(-\frac{1}{i \hbar}\int_{0}^1 |\gamma'_X|^2 d \gamma_\Sigma}{\to}
  \mathbb{C}
  \,.
\end{displaymath}
Notice that $\mathbf{\Pi}(X)\times \mathbf{\Pi}(\Sigma) \simeq \mathbf{\Pi}(X \times \Sigma)$.

\hypertarget{the_quantization}{}\subsubsection*{{The quantization}}\label{the_quantization}

The total action is differential cocycle

\begin{displaymath}
\itexarray{
    \Sigma \times X 
    &\stackrel{}{\to}& Vect
    \\
    \downarrow && \downarrow 
    \\
    \mathbf{\Pi}(\Sigma) \times 
     \mathbf{\Pi}(X) &\stackrel{\exp(S_{kin})tra_\nabla}{\to}& \mathbf{E}\mathbf{B}\mathbb{R} 
      \coprod_{\mathbf{B}U(1)} Vect
  }
  \,.
\end{displaymath}
We want to consider the diagram

\begin{displaymath}
\itexarray{
    \Sigma \times X & \to&  Vect
    \\
    \downarrow & \searrow && \searrow
    \\
    \Sigma && \mathbf{\Pi}(\Sigma)\times \mathbf{\Pi}(X) &\to& \mathbf{E}\mathbf{B}U(1)\coprod_{\mathbf{B}U(1)} Vect
    \\
    & \searrow & \downarrow
    \\
    && \mathbf{\Pi}(\Sigma)
  }
\end{displaymath}
and use it to obtain a differential cocycle on $\Sigma$, by forming something like a lax pullback (``comma object'') of the point inclusion

\begin{displaymath}
\itexarray{
   * &\to& *
   \\
   \downarrow && \downarrow
   \\
   Vect &\to& Vect \coprod_{\mathbf{B}U(1)} \mathbf{E}\mathbf{B}U(1)
 }
 \,,
\end{displaymath}
where the left vertical morphism picks the ground field $\mathbb{C}$, along this cocycle. For the underlying cocycle this is obtained as the ordinary pullback of $\mathbf{E}Vect$ defined as the ordinary pullback

\begin{displaymath}
\itexarray{
    \mathbf{E}Vect &\to& *
    \\
    \downarrow && \downarrow^{\mathrlap{k}}
    \\
    Vect^I &\stackrel{d_1}{\to}& Vect    
  }
  \,,
\end{displaymath}
where $I$ is the [[interval category]], $Vect^I$ the [[functor category]] and $k : * \to Vect$ picks the ground field vector space. Via the remaining map $d_0 : Vect^I \to Vect$ this maps to $Vect$ and then further to $Vect \coprod_{\mathbf{B}U(1)} \mathbf{E}\mathbf{B}U(1)$.

The pullback of

\begin{displaymath}
\itexarray{
   \mathbf{E}Vect &\to& \mathbf{E}Vect
   \\
   \downarrow && \downarrow
   \\
   Vect &\to& Vect \coprod \mathbf{E}\mathbf{B}U(1)
 }
\end{displaymath}
along our differential cocycle, i. e. the pullback of the top part of the diagram

\begin{displaymath}
\itexarray{
    && \mathbf{E}Vect
    \\
    && \downarrow & \searrow
    \\
    \Sigma \times X & \to&  Vect && \mathbf{E}Vect
    \\
    \downarrow & \searrow && \searrow & \downarrow
    \\
    \Sigma && \mathbf{\Pi}(\Sigma)\times \mathbf{\Pi}(X) &\to& \mathbf{E}\mathbf{B}U(1)\coprod_{\mathbf{B}U(1)} Vect
    \\
    & \searrow & \downarrow
    \\
    && \mathbf{\Pi}(\Sigma)
  }
\end{displaymath}
is over $X \times \Sigma$ the total space of the pullback of the underlying vector bundle $E$ on $X$ to $X \times \Sigma$, and over $\mathbf{\Pi}(X \times \Sigma)$ is a groupoid $E_{\mathbf{\Pi}}$

\begin{displaymath}
\itexarray{
    E &\to& E_{\mathbf{\Pi}}
    \\
    \downarrow && \downarrow
    \\
    X\times \Sigma &\to& \mathbf{\Pi}(X \times \Sigma)
    \\
    \downarrow && \downarrow
    \\
    \Sigma &\to& \mathbf{\Pi}(\Sigma) 
  }
  \,.
\end{displaymath}
To see what $E \to E_{\mathbf{\Pi}}$ is like, first notice that we have a [[fibration sequence]]

\begin{displaymath}
\itexarray{
    &&  && V
    \\
    && && \downarrow
    \\
    V &\to& E &\to & V//U(1) &\to& *
    \\
    \downarrow && \downarrow && \downarrow && \downarrow
    \\
    {*} &\stackrel{x}{\to}
    &X &\stackrel{g}{\to}& \mathbf{B}U(1) &\stackrel{\rho}{\to}&
    Vect 
  }
\end{displaymath}
with the bottom right square a lax pullback and everything else [[homotopy pullback]]s. Here $V$ is the vector space that $\rho : \mathbf{B}U(1) \to Vect$ is a representation on (which for the [[electromagnetic field]] will be $\mathbb{C}$ itself).

The groupoid $E_{\mathbf{\Pi}}$ has the following description:

\begin{itemize}%
\item its objects are triples $(x,\sigma, v)$, where $v \in E_{x}$ is a vector in the fiber of $E$ over $x$;


\item its morphisms are triples $(x \stackrel{\gamma}{\to} y, \sigma \to \sigma', v)$ that go from $(x,\sigma,v)$ to $(y, \sigma', \exp(\int_0^1 |...|) \rho(\nabla(\gamma)))(v)$, i.e. from a vector in the fiber over the source to the corresponding vector in the fiber over the target, obtained by evaluating the action functional on the path.



\end{itemize}
To obtain from this a cocycle on $\Sigma$, we proceed as follows: we regard an interval $\sigma := [\sigma_{in}, \sigma_{out}] \in \mathbf{\Pi}(\Sigma)_1$ as a [[cospan]]

\begin{displaymath}
\itexarray{
    \sigma_{in} &\to& \sigma &\leftarrow& \sigma_{out}
    \\
    \downarrow && \downarrow && \downarrow
    \\
    \sigma_{in} &\to& \mathbf{\Pi}(\sigma) &\leftarrow& \sigma_{out}
    \,,
  }
\end{displaymath}
where in the top row we regard these subsets of $\Sigma$ as discrete smooth sub-categories, and in the bottom row form the path $\infty$-groupoids.

Then we take [[section]]s of $\itexarray{E &\to& E_{\mathbf{\Pi}}\\ \downarrow && \downarrow \\ \Sigma &\to& \mathbf{\Pi}(\Sigma)}$ to produce a [[span]] of sections

\begin{displaymath}
\left[
    \itexarray{
      \sigma_{in}
      \\
      \downarrow
      \\
      \sigma_{in} 
    }
    \,,
    \itexarray{
      E
      \\
      \downarrow
      \\
      E_{\mathbf{\Pi}} 
    }
  \right]_\Sigma
  \leftarrow
  \left[
    \itexarray{
      \sigma
      \\
      \downarrow
      \\
      \mathbf{\Pi}(\sigma)
    }
    \,,
    \itexarray{
      E
      \\
      \downarrow
      \\
      E_{\mathbf{\Pi}} 
    }
  \right]_\Sigma  
  \to 
  \left[
    \itexarray{
      \sigma_{out}
      \\
      \downarrow
      \\
      \sigma_{out} 
    }
    \,,
    \itexarray{
      E
      \\
      \downarrow
      \\
      E_{\mathbf{\Pi}} 
    }
  \right]_\Sigma
  \,.
\end{displaymath}
of smooth $\infty$-groupoids.

Consider an $\infty$-groupoid

\begin{displaymath}
\Psi \to
  \left[
    \itexarray{
      \sigma_{in}
      \\
      \downarrow
      \\
      \sigma_{in} 
    }
    \,,
    \itexarray{
      E
      \\
      \downarrow
      \\
      E_{\mathbf{\Pi}} 
    }
  \right]_\Sigma
\end{displaymath}
over the left foot. Under [[groupoid cardinality]], if $\Psi$ is tame, this corresponds to a collection of rational numbers over vectors in fibers of $E$. Under ``degrupoidification'' we may think of this as specifying a section $|\Psi| \in \Gamma(E)$. The above span is supposed to give us the propagation of this state along $\sigma$.

To determine this, consider the special case where $\Psi$ is a ``delta-section'', $* \mapsto (x,\sigma_{in}, v)$ supported on a single vector $v$ in a single fiber $E_x$ over $x$. Then its pull-push through this span yields the $\infty$-groupoid over $E$, which over $(y,\sigma_{out},w)$ consists of the set of paths $\gamma : x \to y$ such that $w = \exp(\int...)\rho(\gamma)(v)$, i.e. such that $w$ is the vector obtained from applying the action to $v$ along this path.

If everything were suitably finite, we could take cardinalities of the result and obtain the familiar path integral (sum)

\begin{displaymath}
\Psi'(y)
  =
  \int_{x \stackrel{\gamma}{\to} y}
  \exp( S_{kin}(\gamma)) tra_\nabla(\gamma) \Psi(x)  
  \,.
\end{displaymath}
[[!redirects exercise in groupoidification -- the path integral]] [[!redirects An Exercise in Groupoidification]]



\end{document}